Shifted<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:mfrac></mml:math>expansions for energy eigenvalues of the Schrödinger equation

Sukhatme, U.; Imbo, Tom D.

doi:10.1103/physrevd.28.418

Cited by 115 publications

(60 citation statements)

References 10 publications

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“…Our rst solvable example is the three-body potential of the form V 1 = p 3f 1 2r 2 " (x 1 + x 2 2x 3 ) (x 1 x 2 ) + cyclic terms # ; (198) which is added to the Calogero potential V C of eq. (196 (199) To see why potentials given by eqs. (196) to (198) …”

Section: Shape Invariance and 3-body Solvable Potentialsmentioning

confidence: 99%

“…A slightly modied, physically motivated approach, called the \shifted large-N method" [196,197,198,199] incorporates exactly known analytic results into 1=N expansions, greatly enhancing their accuracy, simplicity and range of applicability. In this subsection, we will descibe how the rate of convergence of shifted 1=N expansions can be still further improved by using the ideas of SUSY QM [77].…”

Section: Supersymmetry and Large-n Expansionsmentioning

confidence: 99%

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Supersymmetry and quantum mechanics

1995

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In the past ten years, the ideas of supersymmetry have been profitably applied to many nonrelativistic quantum mechanical problems. In particular, there is now a m uch deeper understanding of why certain potentials are analytically solvable and an array o f p o w erful new approximation methods for handling potentials which are not exactly solvable. In this report, we review the theoretical formulation of supersymmetric quantum mechanics and discuss many applications. Exactly solvable potentials can be understood in terms of a few basic ideas which include supersymmetric partner potentials, shape invariance and operator transformations. Familiar solvable potentials all 1 have the property of shape invariance. We describe new exactly solvable shape invariant potentials which include the recently discovered self-similar potentials as a special case. The connection between inverse scattering, isospectral potentials and supersymmetric quantum mechanics is discussed and multi-soliton solutions of the KdV equation are constructed. Approximation methods are also discussed within the framework of supersymmetric quantum mechanics and in particular it is shown that a supersymmetry inspired WKB approximation is exact for a class of shape invariant potentials. Supersymmetry ideas give particularly nice results for the tunneling rate in a double well potential and for improving large N expansions. We also discuss the problem of a charged Dirac particle in an external magnetic eld and other potentials in terms of supersymmetric quantum mechanics. Finally, w e discuss structures more general than supersymmetric quantum mechanics such as parasupersymmetric quantum mechanics in which there is a symmetry between a boson and a para-fermion of order p.

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Section: Shape Invariance and 3-body Solvable Potentialsmentioning

confidence: 99%

Section: Supersymmetry and Large-n Expansionsmentioning

confidence: 99%

Supersymmetry and quantum mechanics

1995

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show abstract

“…A slightly modified, physically motivated approach, called the "shifted large-N method" [196,197,198,199] incorporates exactly known analytic results into 1/N expansions, greatly enhancing their accuracy, simplicity and range of applicability. In this subsection, we will descibe how the rate of convergence of shifted 1/N expansions can be still further improved by using the ideas of SUSY QM [77].…”

Section: Supersymmetry and Large-n Expansionsmentioning

confidence: 99%

Parasupersymmetry in quantum mechanics

Khare

1993

Journal of Mathematical Physics

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In the past ten years, the ideas of supersymmetry have been profitably applied to many nonrelativistic quantum mechanical problems. In particular, there is now a much deeper understanding of why certain potentials are analytically solvable and an array of powerful new approximation methods for handling potentials which are not exactly solvable. In this report, we review the theoretical formulation of supersymmetric quantum mechanics and discuss many applications. Exactly solvable potentials can be understood in terms of a few basic ideas which include supersymmetric partner potentials, shape invariance and operator transformations. Familiar solvable potentials all 1 have the property of shape invariance. We describe new exactly solvable shape invariant potentials which include the recently discovered self-similar potentials as a special case. The connection between inverse scattering, isospectral potentials and supersymmetric quantum mechanics is discussed and multi-soliton solutions of the KdV equation are constructed. Approximation methods are also discussed within the framework of supersymmetric quantum mechanics and in particular it is shown that a supersymmetry inspired WKB approximation is exact for a class of shape invariant potentials. Supersymmetry ideas give particularly nice results for the tunneling rate in a double well potential and for improving large N expansions. We also discuss the problem of a charged Dirac particle in an external magnetic field and other potentials in terms of supersymmetric quantum mechanics. Finally, we discuss structures more general than supersymmetric quantum mechanics such as parasupersymmetric quantum mechanics in which there is a symmetry between a boson and a para-fermion of order p.

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“…Here we mention the standard perturbation RayleighSchrödinger theory ( [1][2]), the quasiclassical or WKB method ( [1][2]), 1/N -expansion ( [3,4]). We will not go into details of these methods and refer readers to the numerous literature (see, for example, [1][2][3][4]).…”

Section: Introductionmentioning

confidence: 99%

Axially symmetric potentials in the oscillator representation

Dineykhan

1997

Z Phys D - Atoms, Molecules and Clusters

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The Wick-ordering method called the Oscillator Representation in the nonrelativistic Schrödinger equation is proposed to calculate the energy spectrum for axially symmetric potentials allowing the existence of a bound state. In particular, the method is applied to calculate the energy spectrum of (2s) states of a hydrogen atom in a uniform magnetic field of an arbitrary strength. In the perturbation (external field) approximation, the energy spectrum of the so-called quadratic and spherical quadratic Zeeman problem and the problem of a hydrogen atom in a generalized van der Waals potential is calculated analytically. The results of the zeroth approximation of oscillator representation are in good agreement with the exact values.

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Shifted1Nexpansions for energy eigenvalues of the Schrödinger equation

Cited by 115 publications

References 10 publications

Supersymmetry and quantum mechanics

Supersymmetry and quantum mechanics

Parasupersymmetry in quantum mechanics

Axially symmetric potentials in the oscillator representation

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